Nuclear

Modelled[M] modelled package — not free public theorem; see doctrine v1.0.3 residual chips on /status.

Hadronic Hagedorn Freeze-Out

The thermodynamic freeze-out temperature of hadronic matter derived as Tc ≈ 205 MeV from the canonical partition function evaluated over the discrete spectrum of coprime torus knots T(p,q).

Last updated: 9 August 2026

Reading this entry

The Hagedorn page reads chemical freeze-out as the heat-capacity peak of coprime torus knots. With the self-linking unit set by π⁵ m_e, the canonical partition function peaks at T_c ≈ 205 MeV, and the page sets that temperature beside the heavy-ion and lattice benchmarks.

What this is used for

Proton and neutron pages share the coprime knots T(2,3) and T(3,2) that enter this sum. Baryogenesis and the spectrum skeleton can take T_c ≈ 205 MeV as the modelled temperature where the heat capacity of those knots peaks.

Walk through

Coprime torus knots T(p, q), with gcd(p, q) = 1, carry the energy E(p, q) = pq · π⁵ m_e ≈ pq · 156.38 MeV, using the unit π⁵ m_e ≈ 156.4 MeV inside the quoted window T_c ≈ 150–200 MeV. The canonical sum is Z(T) = Σ exp(−pq π⁵ m_e / k_B T) over those pairs, and the heat capacity is C_v(T) = [1/(k_B T)²] [⟨E²⟩ − ⟨E⟩²]. The listed low rungs are pq = 1 at 156.4 MeV and pq = 2 at 312.8 MeV. Scanning temperature, C_v peaks at T_c ≈ 205 MeV, which is 2.38×10^12 K. The comparison rows print ALICE at 156.5 ± 1.5 MeV (1.82×10^12 K) and lattice QCD at 155–175 MeV.

Integrity note (Doctrine v1.0.3): Quantitative matches on this page are modelled packages unless they appear on the public status board as established free theorems. Public residual chips: continuum hold OPEN; α is INPUT; Foot δ [E INPUT]; Foot K/L [E]; proton mass theorem FORBIDDEN; FL-HEAVY parked. Soft packages are not finished UFT certificates.

1. Hadronic Phase Transition & Freeze-Out

In ultra-relativistic heavy-ion collisions (LHC at CERN and RHIC at Brookhaven) and in the early universe, matter transitions from a hot plasma of unconfined field excitations into bound hadrons at a characteristic chemical freeze-out temperature .

In the CPT framework, elementary hadrons are closed torus-knot topological structures with coprime winding numbers . By evaluating the canonical partition function over the discrete spectrum of coprime torus knots, we derive a sharp thermodynamic heat capacity peak at , setting the freeze-out scale directly from the knot self-linking unit .

2. Torus-Knot Statistical Mechanics

Each allowed coprime torus knot carries a self-linking energy scale :

Canonical Partition Function & Dimensionless Heat Capacity Peak

3. Freeze-Out Benchmark Comparison

Framework / Experiment	Freeze-Out Temp (MeV)	Temperature (K)	Physical Mechanism
ALICE / LHC Heavy-Ion Data	156.5 ± 1.5 MeV	1.82 × 10¹² K	Chemical Hadronization
Lattice QCD Simulations	155 – 175 MeV	1.95 × 10¹² K	Chiral / Deconfinement Transition
CPT Torus-Knot [M]	~205 MeV	2.38 × 10¹² K	Torus-Knot Heat Capacity Peak

STATUS: [M] MODEL-CONSISTENT DERIVATION

The hadronic freeze-out temperature is a model-consistent result [M] derived from the statistical mechanics of coprime torus knots with self-linking energy scale .

Source Documents

The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.

Documents open in the CPT Research Library — a formatted viewer for all source research files.